I think Isaiah was speaking of the Messiah.
First, there is no scholarly consensus on who Isaiah 9:1-7 (Eng.) is speaking of. Many great OT scholars think that the child is messianic. Brevard S. Childs puts it this way, raising the same point of the child's everlasting reign as you do: "The description of his reign makes it absolutely clear that his role is messianic. There is no end to his rule upon the throne of David, and he will reign with justice and righteousness forever" - Brevard S. Childs, Isaiah, p. 81, emphasis mine.
Second, scholars typically relate Isaiah 9 to Isaiah 7, where the child there is clearly meant to be born in Isaiah's day (cf. 7:16-17). Thus, many scholars take Isaiah 9 as about a figure contemporary to Isaiah. However, in my opinion, the child Immanuel, promised in 7:14, has an initial fulfillment in the child Maher-Shalal-Hash-Baz, the sign of judgement in Isaiah 8:1-4, and an ultimate fulfillment in the messianic child of chapter 9, who comes out of Galilee, and whose reign and Kingship is universal. Once Maher-Shalal-Hash-Baz was born, Isaiah was able to expand upon the already mysterious sign given to Ahaz. Isaiah 8:9-10 already turns to the eschatological future, beyond the Assyrian conflict mentioned in the context, with Immanuel being appealed to against the onslaught of "many nations" (plural) from "all ... distant lands,"
"The valence of the prophecy then reaches in a messianic direction, as Isaiah 8:10 applies it to the eschatological assault of many nations on God’s anointed (see Ps 2; note the Hebrew ʿimmānû ʾĒl in 8:10)" - Stephen L. Cook, John T. Strong, Steven S. Tuell, The Prophets: Introducing Israel's Prophetic Writings (Fortress Press, 2022), p. 99
Isaiah 8:9-10 also seems influenced or akin to the Psalms of Zion, where the nations rally against Israel in the eschatological future (and against God's anointed, as in Psalm 2). This sudden switch from the Assyrian invasion in vv. 6-8 to the eschatological future featuring said "many nations" and "all ... distant lands" in vv. 9-10 is understandable when one realizes that recursion is common for various topics (such as the eschatological future) in Hebrew prophetic literature. Thus, Brevard Childs writes, "Notwithstanding the extraordinary mystery and indeterminacy surrounding the giving of the sign of Immanuel, there are many clear indications that it was understood messianically by the tradents of the Isaianic tradition and shaped in such a way both to clarify and expand the messianic hope for every successive generation of the people of God" - Brevard S. Childs, Isaiah (OTL, 2001), pp. 68-69.
Third, many scholars appeal to the use of perfects in Isaiah 9 to denote a past event. I agree with the past tense translation of the perfects, but that does not necessarily mean the referent is in the past (see e.g., the future reference for מָלְאָ֣ה in Isaiah 11:9). In Hebrew, perfects do not always necessitate a past referent. For example, the author(s) of Isaiah indicate that much if the material was seen in visions (e.g., Isaiah 1:1). If the information in Isaiah 9 was seen in a vision, it could have a past tense translation since the author saw the future in the past through a vision. This is a part of what some scholars call the "prophetic perfect tense," though this ends up being a misnomer, though that's another discussion.
Fifth, scholars do not typically translate the names as Chabad does. H.G.M. Williamson calls this construction "least plausible" (Williamson, Isaiah 6-12, p. 396). Rather, many scholars take the names as theophoric and/or rhetorical hyperbole, similar to Psalm 45. However, many other critical scholars see the inadequacy of comparing Psalm 45 to Isaiah 9, and also see the force of the names in Isaiah 9 such that divinity is implied if they are left as stands. There is has thus been a recent wave of Isaian scholarship that has tried to separate the four names into two clauses, thus resulting in two theophoric names ("wonderful planner [is] the mighty God; the Father of eternity [is] a prince of peace"). Paul D. Wegner for example writes:
"This interpretation would be favoured by: (1) its similarity to the parallel structure in the name Maher-shalal-hash-baz; (2) the translation of sem as one name which the singular form suggests; (3) the Masoretic pointing; and (4) the common pattern in theophoric names" (Wegner 1992: 111).
H.G.M. Williamson agrees with Wegner's interpretation (Williamson, Isaiah 6-12, p. 398), as do scholars like John Goldingay and Gregory Goswell. In response to this, theophoric names are not usually that long-winded. Theophoric names, in contrast to Isaiah 9, normally have two elements, the divine name or title and an accompanying verb, adjective, or noun. Even worse, the "names" Wegner suggests don't even make much sense. They force together incongruous elements. Williamson admits that this reading of the names (and indeed taking them as theophoric in general) is a "minority line of interpretation" on p. 397 (ICC, 2018).